Summary: On OneWay Functions with Optimal Locality
Benny Applebaum Yuval Ishai Eyal Kushilevitz
Computer Science Department, Technion
August 18, 2005
In  it was shown that, under relatively mild assumptions, there exist oneway functions (OWFs) in NC 0
namely ones in which each bit of the output depends on just 4 input bits. This result is not far from optimal as it
is impossible to implement OWF in NC 0
2 . The gap was partially closed in  by showing that the existence of a
OWF in NC 0
3 is implied by the intractability of decoding a random linear code.
In this note we provide further evidence for the existence of OWF in NC 0
3 . We construct such a OWF based
on the existence of a OWF that enjoys a certain strong ``robustness'' property. Specifically, we require that the
adversary cannot invert f even if it is given, in addition to the output y = f(x), all bits of x influencing a randomly
chosen subset of the bits of y. We show that a variant of a OWF candidate suggested by Goldreich  satisfies
this property, assuming that it is indeed a OWF.